Question

The number of real solutions of the equation $sin\left(2^x\right)=5^x+5^{-x}$ is

• A. 0
• B. 1
• C. 2
• D. infinitely many

Answer 1

The correct option is A 0

Given Equation is $\sin \left(2^x\right)=5^x+5^{-x}$

$AM\ge GM$

$⟹\frac{5^x+5^{-x}}{2}\ge \sqrt{5^x\times 5^{-x}}$

$⟹5^x+5^{-x}\ge 2$

But $\sin \left(2^x\right)\in [-1,1]$

So the equality does not exist for any value of $x$

The number of real solutions of the given equation is $0$.